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On a problem of Bernik, Kleinbock and Margulis. (English) Zbl 1268.11098
The author considers the system of inequalities \[ | P(x) | < \Psi_1(H(P)), \quad | P'(x) | < \Psi_2(H(P)), \] where \(P\) is an integer polynomial, \(H(P)\) is the maximum among the absolute values of the coefficients of \(P\) and the \(\Psi_i\) are real, positive functions. The set \({\mathcal P}_n(\Psi_1, \Psi_2)\) consists of the numbers \(x \in [-1/2, 1/2]\) for which this system of inequalities is satisfied for infinitely many integer polynomials \(P\) of degree at most \(n\).
Under various technical restrictions on the functions \(\Psi_1\) and \(\Psi_2\), the Lebesgue measure of the set \({\mathcal P}_n(\Psi_1, \Psi_2)\) is calculated. With the technical assumptions ignored, the set is null or full according to the convergence or divergence of the series \(\sum_{h=1}^\infty h^{n-2} \Psi_1(h) \Psi_2(h)\), a result very much like the classical Khintchine theorem. The paper is a first step towards establishing a Khintchine type theorem in this context, although there is still a long way to go due to the technical conditions. The proof uses Sprindzhuk’s method of essential and inessential domains along with more recent results.
MSC:
11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
11J13 Simultaneous homogeneous approximation, linear forms
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